Open submanifold - Knowlpedia

Let $M$ be a smooth manifold. An open submanifold of $M$ is an open subset $U\subseteq M$ equipped with the smooth structure generated by the restricted charts $(U\cap V,\varphi|_{U\cap V})$ , where $(V,\varphi)$ ranges over the charts of $M$ .

The inclusion $U\hookrightarrow M$ is a smooth open embedding, and $\dim U=\dim M$ .

Universal property for smooth maps

A map $f:N\to U$ is smooth exactly when its composite with the inclusion $U\hookrightarrow M$ is smooth. Thus the induced structure is uniquely characterized by the requirement that maps into $U$ can be tested after viewing them as maps into $M$ .

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. DOI record. Relevant: open submanifolds and smooth maps.